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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > undif5 | Structured version Visualization version GIF version |
Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.) |
Ref | Expression |
---|---|
undif5 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difun2 4370 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) | |
2 | disjdif2 4369 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) | |
3 | 1, 2 | syl5eq 2806 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∖ cdif 3851 ∪ cun 3852 ∩ cin 3853 ∅c0 4221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 |
This theorem is referenced by: fressupp 30531 |
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